Animate Branch Growth
You can achieve a simplistic tree growth animation using the Bevel Factor property of the curve.



You might have to separate your tree into several discreet branch levels, and for a proper realistic effect it will probably require a lot of manual work.

Create a Sapling tree, enter Edit Mode select the whole main trunk and separate it with P.

Then select one of the branch tips (radius $0$), Shift+G Select Grouped > Radius then Ctrl+L to select all linked vertex. Press P again to separate that whole branch level.

Repeat for every branch level you wish to animate independently.



You can then animate the individual Bevel Factor properties on each curve object to achieve a simplistic growth animation by pressing keyframing the property with i while hovering it.

Separate into further objects for more detailed growth, for example by height levels, so lower branches can start growing before the upper ones.